BLED WORKSHOPS
IN PHYSICS
VOL. 13, NO. 1
p. 1
Proceedings of the Mini-Workshop
Hadronic Resonances
Bled, Slovenia, July 1 - 8, 2012
The Road to Extraction of S-Matrix Poles from
Experimental Data ⋆
Saša Ceci, Milorad Korolija, and Branimir Zauner
Rudjer Bosković Institute, Bijenička 54, HR-10000 Zagreb, Croatia
Abstract. By separating data points close to a resonance into intervals, and fitting all pos-
sible intervals to a simple pole with constant coherently added background, we obtained
a substantial number of convergent fits. After a chosen set of statistical constraints was
imposed, we calculated the average of a resonance pole position from the statistically ac-
ceptable results. We used this method to find pole positions of Z boson.
Breit-Wigner (BW) parameters are often used for the description of unstable par-
ticles (see e.g., Review of Particle Physics [1]), although shortcomings of such choice
have been pointed out on numerous occasions. For example, Sirlin showed that
the BWparameters of the Z boson were gauge dependent [2]. To resolve this issue
he redefined BWparameters, but also suggested usage of the S-matrix poles as an
alternative, since poles are fundamental properties of the S-matrix and therefore
gauge independent by definition. In a somewhat different study, Höhler advo-
cated using S-matrix poles for characterization of nucleon resonances [3] in order
to reduce confusion that arises when different definitions of BW parameters are
used [4]. However, loosely defined [5] BW parameters of mesons and baryons are
still being extracted from experimental analyses, compared among themselves
[1], and used as input to QCD-inspired quark models [6] and as experiment-to-
theory matching points for lattice QCD [7].
Our group has been very interested in reducing human and model depen-
dence from resonance parameters’ extraction procedures (from scattering matri-
ces). We developed the regularization method for pole extraction from S-matrix
elements [8]. Its main disadvantage is that it needs very dense data, one that is at-
tainable only after an energy dependent partial-wave analysis. The other method
was the K-matrix pole extraction method [9] which needed the whole unitary S-
matrix to begin with, making it impossible to use on any single reaction. Both
of those methods were purely mathematical, and the only assumption were that
there is a pole in the complex energy plane of an S-matrix. We had no physical
input into our procedures. Therefore, we proclaimed these procedures model-
independent. The only thing missing, was a method which could be applied di-
rectly to the experimental data, e.g., total cross sections.
In this proceeding, we illustrate a method for model-independent extraction
of S-matrix pole positions directly from the data.
⋆ Talk delivered by S. Ceci
2 Saša Ceci, Milorad Korolija, and Branimir Zauner
The first step in devising a method for extraction of the pole parameters from
the experimental data is to set up an appropriate parameterization. The parame-
terization presented here is based on the assumption that close to a resonance, the
T matrix will be well described with a simple pole and a constant background.
The similar assumption was used in Höhler’s speed plot technique [3]. The speed
plot is a method used for the pole parameter extraction from the known scat-
tering amplitudes. It is based on calculating the first order energy derivative of
the scattering amplitude, with the key assumption that the first derivative of the
background is negligible.
The T matrix with a single pole and constant background term is given by
T(W) = rp
Γp/2
Mp −W − i Γp/2
+ bp, (1)
whereW is center-of-mass energy, rb and bb are complex, whileMp and Γp are
real numbers. Total cross section is then proportional to |T |2/q2, where q is the
initial center-of-mass momentum. Equation (1), as well as other similar forms
(see e.g. [1]), are standardly called Breit-Wigner parameterizations, which can be
somewhat misleading since Mp and Γp are generally not Breit-Wigner, but pole
parameters (hence the index p). The square of the T matrix defined in Eq. (1) is
given by
|T(W)|2 = T2
∞
(W −Mz)
2 + Γ2z /4
(W −Mp)2 + Γ2p/4
, (2)
where, for convenience, we simplified the numerator by combining the old pa-
rameters into three new real-valued ones: T∞, Mz, and Γz. Pole parametersMp
and Γp are retained in the denominator.
With such a simple parameterization, it is crucial to use only data points
close to the resonance peak. To avoid picking and choosing the appropriate data
points by ourselves, we analyzed the data from a wider range around the reso-
nance peak, and fitted localy the parameterization (2) to each set of seven succes-
sive data points (seven data points is minimum for our five-parameter fit). Then
we increased the number of data points in the sets to eight and fitted again. We
continued increasing the number of data points in sets until we fitted the whole
chosen range. Such procedure allowed different background term for each fit,
which is much closer to reality than assuming a single constant background term
for the whole chosen data set (see e.g. discussion on the problems with speed plot
in Ref. [8]). In the end, we imposed a series of statistical constraints to all fits to
distinguish the good ones. The whole analysis was done in Wolfram Mathemat-
ica 8 using NonlinearModelFit routine [11].
Having defined the fitting strategy, we tested the method by applying it to
the case of the Z boson. The data set we used is from the PDG compilation [1],
and shown in Fig. 1. Extracted pole masses are shown in the same figure: filled
histogram bins show pole masses from the good fits, while the empty histogram
bins are stacked to the solid ones to show masses obtained in the discarded fits.
Height of the pole-mass histogram in Fig. 1 is scaled for convenience.
Extracted S-matrix pole mass and width of Z boson are given in Table 1. The
polemasses are in excelent agreement, while the pole widths are reasonably close.
The Road to Extraction of S-Matrix Poles from Experimental Data 3
88 89 90 91 92 93 94 95
0
5
10
15
20
25
30
35
40
W / GeV
R
/1
02
Z
88 89 90 91 92 93 94 95
0
1
2
3
4
5
6
Mp / GeV
G
p
/G
eV
Z
Fig. 1. [Upper figure] PDG compilation of Z data [1] and histogram of obtained pole
masses. Line is the fit result with the lowest reduced χ2 (just for illustration). Dark (red
online) colored histogram bins are filled with statistically preferred results. [Lower figure]
Pole masses vs. pole widths. Dark (red online) circles show statistically preferred results
we use for averages.
It is important to stress that the difference between the pole and BWmass of the Z
boson is fundamental and statistically significant. Distribution of discarded and
good results is shown in the lower part of Fig. 1.
Table 1. Pole parameters of Z obtained in this work. PDG values of pole and BW parame-
ters are given for comparison.
Pole Pole PDG [1] BW PDG [1]
M/MeV 91159 ± 8 91162 ± 2 91188 ± 2
Γ/MeV 2484 ± 10 2494 ± 2 2495 ± 2
4 Saša Ceci, Milorad Korolija, and Branimir Zauner
In conclusion, we have illustrated here a model-independent method for ex-
traction of resonance pole parameters from total cross sections and partial waves.
Very good estimates for Z boson pole position were obtained.
We are today witnessing the dawn of ab-initio calculations in low-energy
QCD. In order to compare theoretical predictionswith experimentally determined
resonance states, we need first to establish proper point of comparison. We hope
that our method, once it becomes fully operational, will help connecting experi-
ment and theory.
References
1. J. Beringer et al., (Particle Data Group), Phys. Rev. D86, 010001 (2012)
2. A. Sirlin, Phys. Rev. Lett. 67, 2127 (1991).
3. G. Höhler, πNNewsletter 9, 1 (1993).
4. R. E. Cutkosky et al., Phys. Rev. D20, 2839 (1979); D. M. Manley and E. M. Saleski,
Phys. Rev. D45, 4002 (1992).
5. G. Höhler, “A pole-emic” in Review of particle properties, D. E. Groom et al.
Eur. Phys. J. C15, 1 (2000).
6. S. Capstick and W. Roberts Prog. Part. Nucl. Phys. 45, S241-S331 (2000); T. Melde,
W. Plessas, and B. Sengl, Phys. Rev. D77, 114002 (2008).
7. S. Dürr, et al., Science 322, 1224 (2008).
8. S. Ceci et al., J. Stahov, A. Švarc, S. Watson, and B. Zauner, Phys. Rev. D77, 116007
(2008).
9. S. Ceci, A. Švarc, B. Zauner, D. M. Manley, and
S. Capstick, Phys. Lett. B659, 228 (2008).
10. http://gwdac.phys.gwu.edu/analysis/pin analysis.html Current PWA solution (June, 2010).
11. http://reference.wolfram.com/mathematica/ref/
NonlinearModelFit.html